Closed Foldy-Wouthuysen transformations for fermions moving in gauge-invariant time-dependent electromagnetic fields

V.P.Neznamov

arXiv:2509.00021·hep-th·September 3, 2025

Closed Foldy-Wouthuysen transformations for fermions moving in gauge-invariant time-dependent electromagnetic fields

V.P.Neznamov

PDF

TL;DR

This paper generalizes the Foldy-Wouthuysen transformation to fermions in gauge-invariant, time-dependent electromagnetic fields, providing closed-form energy operators and linking to the Feynman-Gell-Mann representation.

Contribution

It introduces a method to obtain closed-form energy operators for fermions in dynamic electromagnetic fields, extending previous static field results.

Findings

01

Derived closed-form energy operators in time-dependent fields

02

Established connection with Feynman-Gell-Mann representation

03

Extended previous static field results to dynamic scenarios

Abstract

Previously, we obtained closed expressions for energy operators in the Foldy-Wouthuysen representation in the presence of static electric fields. In this case, we also established a connection between the Foldy-Wouthuysen representation and the Feynman-Gell-Mann representation. In this work, we generalize these results for the case of fermions moving in time-dependent gauge-invariant electromagnetic fields.

Equations98

p^{0} ψ_{D} (x, t) = H_{D} (t) ψ_{D} (x, t) = (απ (x, t) + β m + e A^{0} (x, t)) ψ_{D} (x, t) .

p^{0} ψ_{D} (x, t) = H_{D} (t) ψ_{D} (x, t) = (απ (x, t) + β m + e A^{0} (x, t)) ψ_{D} (x, t) .

\alpha^{i}=\left({\begin{array}[]{l}0\;\;\,\sigma^{i}{\kern 1.0pt}\\ \sigma^{i}\;\;0\\ \end{array}}\right),\;\,\,\beta=\gamma^{0}=\left({\begin{array}[]{l}I\;\;\,\,\,\,\,{\kern 1.0pt}0\\ 0\;\,-I\\ \end{array}}\right),\;\,\,\gamma_{5}=\left({\begin{array}[]{l}0\;\;\,{\kern 1.0pt}I\\ I\;\;\;0\\ \end{array}}\right),\;\,\,\gamma^{i}=\gamma^{0}\alpha^{i}.

\alpha^{i}=\left({\begin{array}[]{l}0\;\;\,\sigma^{i}{\kern 1.0pt}\\ \sigma^{i}\;\;0\\ \end{array}}\right),\;\,\,\beta=\gamma^{0}=\left({\begin{array}[]{l}I\;\;\,\,\,\,\,{\kern 1.0pt}0\\ 0\;\,-I\\ \end{array}}\right),\;\,\,\gamma_{5}=\left({\begin{array}[]{l}0\;\;\,{\kern 1.0pt}I\\ I\;\;\;0\\ \end{array}}\right),\;\,\,\gamma^{i}=\gamma^{0}\alpha^{i}.

\begin{array}[]{l}\alpha_{c}^{i}=S\alpha^{i}S^{-1}=\left({\begin{array}[]{l}\sigma^{i}\;\;\,\,\,\,\,\,0{\kern 1.0pt}\\ 0\;\;-\;\sigma^{i}\\ \end{array}}\right),\;\,\beta_{c}=\gamma_{c}^{0}=S\beta S^{-1}=\left({\begin{array}[]{l}0\;\;\,{\kern 1.0pt}I\\ I\;\,\,0\\ \end{array}}\right),\\ \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\gamma_{c}^{5}=S\gamma^{5}S^{-1}=\left({\begin{array}[]{l}I\;\;\,\,\,\,{\kern 1.0pt}0\\ 0\;-I\\ \end{array}}\right),\;\gamma_{c}^{i}=\gamma_{c}^{0}\alpha_{c}^{i}.\\ \end{array}

\begin{array}[]{l}\alpha_{c}^{i}=S\alpha^{i}S^{-1}=\left({\begin{array}[]{l}\sigma^{i}\;\;\,\,\,\,\,\,0{\kern 1.0pt}\\ 0\;\;-\;\sigma^{i}\\ \end{array}}\right),\;\,\beta_{c}=\gamma_{c}^{0}=S\beta S^{-1}=\left({\begin{array}[]{l}0\;\;\,{\kern 1.0pt}I\\ I\;\,\,0\\ \end{array}}\right),\\ \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\gamma_{c}^{5}=S\gamma^{5}S^{-1}=\left({\begin{array}[]{l}I\;\;\,\,\,\,{\kern 1.0pt}0\\ 0\;-I\\ \end{array}}\right),\;\gamma_{c}^{i}=\gamma_{c}^{0}\alpha_{c}^{i}.\\ \end{array}

S=\frac{1}{\sqrt{2}}\left({{\begin{array}[]{*{20}c}I\hfill&\,\,\,\,I\hfill\\ I\hfill&{-I\hfill}\\ \end{array}}}\right).

S=\frac{1}{\sqrt{2}}\left({{\begin{array}[]{*{20}c}I\hfill&\,\,\,\,I\hfill\\ I\hfill&{-I\hfill}\\ \end{array}}}\right).

p^{0} ψ_{c} (x, t) = H_{c} (t) ψ_{c} (x, t) = (α_{c} π (x, t) + β_{c} m + e A^{0} (x, t)) ψ_{c} (x, t) .

p^{0} ψ_{c} (x, t) = H_{c} (t) ψ_{c} (x, t) = (α_{c} π (x, t) + β_{c} m + e A^{0} (x, t)) ψ_{c} (x, t) .

ψ_{c} (x, t) = S ψ_{D} (x, t),

ψ_{c} (x, t) = S ψ_{D} (x, t),

H_{c} (t) = S H_{D} S^{- 1} .

H_{c} (t) = S H_{D} S^{- 1} .

\psi_{c}({\rm{\bf x}},t)=\left({{\begin{array}[]{*{20}c}{\varphi_{c}({\rm{\bf x}},t)\hfill}\\ {\chi_{c}({\rm{\bf x}},t)\hfill}\\ \end{array}}}\right).

\psi_{c}({\rm{\bf x}},t)=\left({{\begin{array}[]{*{20}c}{\varphi_{c}({\rm{\bf x}},t)\hfill}\\ {\chi_{c}({\rm{\bf x}},t)\hfill}\\ \end{array}}}\right).

p^{0} φ_{c} (x, t) = σπ (x, t) φ_{c} (x, t) + m χ_{c} (x, t) + e A^{0} (x, t) φ_{c} (x, t),

p^{0} φ_{c} (x, t) = σπ (x, t) φ_{c} (x, t) + m χ_{c} (x, t) + e A^{0} (x, t) φ_{c} (x, t),

p^{0} χ_{c} (x, t) = - σπ (x, t) χ_{c} (x, t) + m φ_{c} (x, t) + e A^{0} (x, t) χ_{c} (x, t),

p^{0} χ_{c} (x, t) = - σπ (x, t) χ_{c} (x, t) + m φ_{c} (x, t) + e A^{0} (x, t) χ_{c} (x, t),

χ_{c} (x, t) = \frac{m}{p ^{0} + σπ ( x , t ) - e A ^{0} ( x , t )} φ_{c} (x, t),

χ_{c} (x, t) = \frac{m}{p ^{0} + σπ ( x , t ) - e A ^{0} ( x , t )} φ_{c} (x, t),

φ_{c} (x, t) = \frac{m}{p ^{0} - σπ ( x , t ) - e A ^{0} ( x , t )} χ_{c} (x, t) .

φ_{c} (x, t) = \frac{m}{p ^{0} - σπ ( x , t ) - e A ^{0} ( x , t )} χ_{c} (x, t) .

\psi_{c}^{\left(+\right)}({\rm{\bf x}},t)=A^{\left(+\right)}\left({{\begin{array}[]{*{20}c}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,{\varphi_{c}({\rm{\bf x}},t)\hfill}\\[10.0pt] {\dfrac{m}{p^{0}+{\rm{\bm{\sigma\pi}}}\left({{\rm{\bf x}},t}\right)-eA^{0}\left({{\rm{\bf x}},t}\right)}\varphi_{c}({\rm{\bf x}},t)\hfill}\\ \end{array}}}\right),

\psi_{c}^{\left(+\right)}({\rm{\bf x}},t)=A^{\left(+\right)}\left({{\begin{array}[]{*{20}c}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,{\varphi_{c}({\rm{\bf x}},t)\hfill}\\[10.0pt] {\dfrac{m}{p^{0}+{\rm{\bm{\sigma\pi}}}\left({{\rm{\bf x}},t}\right)-eA^{0}\left({{\rm{\bf x}},t}\right)}\varphi_{c}({\rm{\bf x}},t)\hfill}\\ \end{array}}}\right),

\psi_{c}^{\left(-\right)}({\rm{\bf x}},t)=A^{\left(-\right)}\left({{\begin{array}[]{*{20}c}{-\dfrac{m}{-p^{0}+{\rm{\bm{\sigma\pi}}}\left({{\rm{\bf x}},t}\right)+eA^{0}\left({{\rm{\bf x}},t}\right)}\chi_{c}({\rm{\bf x}},t)\hfill}\\[10.0pt] \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,{\chi_{c}({\rm{\bf x}},t)\hfill}\\ \end{array}}}\right).

\psi_{c}^{\left(-\right)}({\rm{\bf x}},t)=A^{\left(-\right)}\left({{\begin{array}[]{*{20}c}{-\dfrac{m}{-p^{0}+{\rm{\bm{\sigma\pi}}}\left({{\rm{\bf x}},t}\right)+eA^{0}\left({{\rm{\bf x}},t}\right)}\chi_{c}({\rm{\bf x}},t)\hfill}\\[10.0pt] \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,{\chi_{c}({\rm{\bf x}},t)\hfill}\\ \end{array}}}\right).

ψ_{c}^{(+)} (x, t)^{†} ψ_{c}^{(+)} (x, t) = φ_{c}^{†} (x, t) φ_{c} (x, t),

ψ_{c}^{(+)} (x, t)^{†} ψ_{c}^{(+)} (x, t) = φ_{c}^{†} (x, t) φ_{c} (x, t),

ψ_{c}^{(-)} (x, t)^{†} ψ_{c}^{(-)} (x, t) = χ_{c}^{†} (x, t) χ_{c} (x, t) .

ψ_{c}^{(-)} (x, t)^{†} ψ_{c}^{(-)} (x, t) = χ_{c}^{†} (x, t) χ_{c} (x, t) .

A^{(+)} = (1 + \frac{m ^{2}}{( p ^{0} + σπ ( x , t ) - e A ^{0} ( x , t ) ) ^{2}})^{- 1 / 12 2},

A^{(+)} = (1 + \frac{m ^{2}}{( p ^{0} + σπ ( x , t ) - e A ^{0} ( x , t ) ) ^{2}})^{- 1 / 12 2},

A^{(-)} = (1 + \frac{m ^{2}}{( - p ^{0} + σπ ( x , t ) + e A ^{0} ( x , t ) ) ^{2}})^{- 1 / 12 2} .

A^{(-)} = (1 + \frac{m ^{2}}{( - p ^{0} + σπ ( x , t ) + e A ^{0} ( x , t ) ) ^{2}})^{- 1 / 12 2} .

\psi_{CFW}^{\left(+\right)}({\rm{\bf x}},t)=U_{CFW}^{\left(+\right)}({\rm{\bf x}},t)\psi_{c}^{\left(+\right)}({\rm{\bf x}},t)=\left({{\begin{array}[]{*{20}c}{\varphi_{c}({\rm{\bf x}},t)\hfill}\\ \,\,\,\,\,\,\,\,0\hfill\\ \end{array}}}\right),

\psi_{CFW}^{\left(+\right)}({\rm{\bf x}},t)=U_{CFW}^{\left(+\right)}({\rm{\bf x}},t)\psi_{c}^{\left(+\right)}({\rm{\bf x}},t)=\left({{\begin{array}[]{*{20}c}{\varphi_{c}({\rm{\bf x}},t)\hfill}\\ \,\,\,\,\,\,\,\,0\hfill\\ \end{array}}}\right),

\psi_{CFW}^{\left(-\right)}({\rm{\bf x}},t)=U_{CFW}^{\left(-\right)}({\rm{\bf x}},t)\psi_{c}^{\left(-\right)}({\rm{\bf x}},t)=\left({{\begin{array}[]{*{20}c}\,\,\,\,\,\,\,\,0\hfill\\ {\chi_{c}({\rm{\bf x}},t)\hfill}\\ \end{array}}}\right).

\psi_{CFW}^{\left(-\right)}({\rm{\bf x}},t)=U_{CFW}^{\left(-\right)}({\rm{\bf x}},t)\psi_{c}^{\left(-\right)}({\rm{\bf x}},t)=\left({{\begin{array}[]{*{20}c}\,\,\,\,\,\,\,\,0\hfill\\ {\chi_{c}({\rm{\bf x}},t)\hfill}\\ \end{array}}}\right).

U_{C F W}^{(+)} = A^{(+)} (1 + \frac{m}{p ^{0} + σπ ( x , t ) - e A ^{0} ( x , t )} γ_{c}^{5} β_{c}),

U_{C F W}^{(+)} = A^{(+)} (1 + \frac{m}{p ^{0} + σπ ( x , t ) - e A ^{0} ( x , t )} γ_{c}^{5} β_{c}),

U_{C F W}^{(-)} = A^{(-)} (1 + \frac{m}{- p ^{0} + σπ ( x , t ) + e A ^{0} ( x , t )} γ_{c}^{5} β_{c}) .

U_{C F W}^{(-)} = A^{(-)} (1 + \frac{m}{- p ^{0} + σπ ( x , t ) + e A ^{0} ( x , t )} γ_{c}^{5} β_{c}) .

H_{C F W}^{(+)} (t) = U_{C F W}^{(+)} (t) H_{c} (t) U_{C F W}^{(+)} (t)^{†} - i U_{C F W}^{(+)} (t) \frac{\partial}{\partial t} (U_{C F W}^{(+)} (t)^{†}),

H_{C F W}^{(+)} (t) = U_{C F W}^{(+)} (t) H_{c} (t) U_{C F W}^{(+)} (t)^{†} - i U_{C F W}^{(+)} (t) \frac{\partial}{\partial t} (U_{C F W}^{(+)} (t)^{†}),

H_{C F W}^{(-)} (t) = U_{C F W}^{(-)} (t) H_{c} (t) U_{C F W}^{(-)} (t)^{†} - i U_{C F W}^{(-)} (t) \frac{\partial}{\partial t} (U_{C F W}^{(-)} (t)^{†}) .

H_{C F W}^{(-)} (t) = U_{C F W}^{(-)} (t) H_{c} (t) U_{C F W}^{(-)} (t)^{†} - i U_{C F W}^{(-)} (t) \frac{\partial}{\partial t} (U_{C F W}^{(-)} (t)^{†}) .

γ_{c}^{5} ψ_{C F W}^{(+)} (t) = ψ_{C F W}^{(+)} (t) .

γ_{c}^{5} ψ_{C F W}^{(+)} (t) = ψ_{C F W}^{(+)} (t) .

γ_{c}^{5} ψ_{C F W}^{(-)} (t) = - ψ_{C F W}^{(-)} (t) .

γ_{c}^{5} ψ_{C F W}^{(-)} (t) = - ψ_{C F W}^{(-)} (t) .

H_{C F W}^{(+)} (t) = (H_{C F W}^{(+)} (t))_{e v e n} + (H_{C F W}^{(+)} (t))_{o dd},

H_{C F W}^{(+)} (t) = (H_{C F W}^{(+)} (t))_{e v e n} + (H_{C F W}^{(+)} (t))_{o dd},

H_{C F W}^{(-)} (t) = (H_{C F W}^{(-)} (t))_{e v e n} + (H_{C F W}^{(-)} (t))_{o dd} .

H_{C F W}^{(-)} (t) = (H_{C F W}^{(-)} (t))_{e v e n} + (H_{C F W}^{(-)} (t))_{o dd} .

(H_{C F W}^{(+)} (t))_{o dd} ψ_{C F W}^{(+)} (x, t) = 0,

(H_{C F W}^{(+)} (t))_{o dd} ψ_{C F W}^{(+)} (x, t) = 0,

(H_{C F W}^{(-)} (t))_{o dd} ψ_{C F W}^{(-)} (x, t) = 0.

(H_{C F W}^{(-)} (t))_{o dd} ψ_{C F W}^{(-)} (x, t) = 0.

Peer Reviews

No public reviews on file for this paper yet. If you reviewed it on a platform where reviews are public (OpenReview, ICLR, NeurIPS, ICML), you can paste yours below so the community can read it here.

Videos

No videos yet. Explain this paper in a talk, walkthrough, or lecture? Add one.

Full text

CLOSED FOLDY-WOUTHUYSEN TRANSFORMATION FOR FERMIONS MOVING IN GAUGE-INVARIANT TIME-DEPENDENT ELECTROMAGNETIC FIELDS

V. P. Neznamov***[email protected], [email protected]

Russian Federal Nuclear Center–All-Russian Research Institute of Experimental Physics*, Mira pr., 37, Sarov, 607188, Russia*

Abstract

Previously, we obtained closed expressions for energy operators in the Foldy-Wouthuysen representation in the presence of static electric fields. In this case, we also established a connection between the Foldy-Wouthuysen representation and the Feynman-Gell-Mann representation. In this work, we generalize these results for the case of fermions moving in time-dependent gauge-invariant electromagnetic fields..

*Keywords: Foldy-Wouthuysen representation, Feynman-Gell-Mann representation, gauge-invariant electromagnetic fields, chiral representation of Dirac matrices.

PACS numbers: 12.20.Ds

1 Introduction

Previously [1], we obtained closed expressions for the energy operators of an electron moving in electrostatic fields in the chiral Foldy-Wouthuysen (CFW) representation [2]. The transition to the CFW representation was implemented using Dirac matrices in the chiral representation.

Closed expressions for Hamiltonians or energy operators are necessary in the presence of strong electromagnetic fields. In this case, perturbation theory is not practical, and we are in the realm of non-perturbative quantum electrodynamics.

In this work, we generalize the results of Ref. [1] for the case of electrons moving in time-dependent gauge-invariant electromagnetic fields.

In Sec. 2 of the article, the Dirac equation in a dynamic gauge-invariant electromagnetic field is considered using Dirac matrices in the standard and chiral representations. In Sec. 3, closed Foldy-Wouthuysen transformations are performed with Dirac matrices in the chiral representation. In Sec. 4, the fermion equations in the Foldy-Wouthuysen representation in the presence of dynamic electromagnetic fields are obtained. A direct connection between the Foldy-Wouthuysen and Feynman-Gell-Mann representations is demonstrated. The Conclusion summarizes the main results of our work.

2 Dirac equation for an electron in a Time-Dependent Gauge-Invariant Electromagnetic Field

The Dirac equation in a time-dependent electromagnetic field has the form

[TABLE]

Hereinafter, we use the system of units of $\hslash=c=1$ ; $H_{D}(t)$ is the Dirac Hamiltonian; $\psi_{D}({\rm{\bf x}},t)$ is the bispinor wave function; $p^{\mu}=i\dfrac{\partial}{\partial x_{\mu}};\,\,\,\mu=0,1,2,3$ ; ${\rm{\bm{\pi}}}({\rm{\bf x}},t)={\rm{\bf p}}-e{\rm{\bf A}}({\rm{\bf x}},t)$ ; $A^{\mu}({\rm{\bf x}},t)$ is the electromagnetic four-potential; $\alpha^{i},\,\,\beta$ are four-dimensional Dirac matrices $\left({i=1,2,3}\right)$ .

The Dirac matrices in the standard representation have the form

[TABLE]

Here, $\sigma^{i}$ are the two-dimensional Pauli matrices, $I$ is the two-dimensional identity matrix.

In modern gauge field theories, and in particular in the Standard Model, the chiral representation of Dirac matrices is widely used

[TABLE]

In Eq. (3), the unitary transformation matrix $S$ has the form

[TABLE]

The Dirac equation (1) with matrices in the chiral representation (3) has the form

[TABLE]

Here,

[TABLE]

Representation (3) in Eq. (5) allows us to obtain closed expressions for Foldy-Wouthuysen transformations in the presence of time-dependent gauge-invariant electromagnetic fields.

The bispinor (6) can be presented as a column of two spinors

[TABLE]

From Eq. (5) with the representation (3), the following equalities can be obtained

[TABLE]

Taking into account (11) and (12), the bispinor (8 ) can be written using only the spinor $\varphi_{c}\left({{\rm{\bf x}},t}\right)$ or only the spinor $\chi_{c}\left({{\rm{\bf x}},t}\right)$

[TABLE]

The normalized operators $A^{\left(+\right)}$ and $A^{\left(-\right)}$ are found from the conditions

[TABLE]

Therefore,

[TABLE]

In the equalities (13), (14) and (17), (18), the numerators of the expressions commute with the corresponding denominators. This allows us to obtain closed expressions of Foldy-Wouthuysen transformations in the presence of dynamic (time-dependent) electromagnetic fields.

3 Closed Foldy-Wouthuysen transformations with Dirac matrices in the chiral representation

Let us recall two conditions of the transition to the Foldy-Wouthuysen representation (see, for instance, Ref. [3]):

The Hamiltonian and fermion equations in the FW representation are diagonal relative to the upper and lower spinors of the transformed wave function $\psi_{FW}({\rm{\bf x}},t)$ . 2. 2.

In the Foldy-Wouthuysen transformations (including CFW transformations in presence of dynamic electromagnetic fields), the reduction conditions for the wave function must be satisfied.

In our case †††Below, the wave functions are normalized by unitary probability in the box of volume $V$ . For brevity, in our expressions for wave functions, there are no multipliers $1\mathord{\left/{\vphantom{1{\sqrt{V}}}}\right.\kern-1.2pt}{\sqrt{V}}$ .,

[TABLE]

The equalities (19), (20) and (13), (14) allow us to define the unitary operators $U_{CFW}^{\left(+\right)\,\,\,\,\,{\dagger}}=\left({U_{CFW}^{\left(+\right)}}\right)^{-1},\,\,\,U_{CFW}^{\left(-\right)\,\,\,\,{\dagger}}=\left({U_{CFW}^{\left(-\right)}}\right)^{-1}$ :

[TABLE]

4 Fermion equations in the Foldy-Wouthuysen representation in the presence of dynamic electromagnetic fields

The time-dependent Hamiltonians in the FW representation have the form

[TABLE]

The second terms in (23) and (24) are absent in case of static electromagnetic fields. $H_{CFW}^{\left(+\right)}\left(t\right)$ acts on the function (19) with

[TABLE]

$H_{CFW}^{\left(-\right)}\left(t\right)$ acts on function (20) with

[TABLE]

The Hamiltonians (23) and (24) can be written as the sum of even and odd parts

[TABLE]

Here, $\left({H_{CFW}^{\left(+\right)}\left(t\right)}\right)_{odd},\,\,\,\left({H_{CFW}^{\left(-\right)}\left(t\right)}\right)_{odd}$ are the parts of the Hamiltonians with the matrix $\beta_{c}$ , which mixes the upper and lower spinors of the wave function.

In the Foldy-Wouthuysen representation, according to condition 1 in Sec. 3,

[TABLE]

When (29) and (30) are satisfied, the Hamiltonians are

[TABLE]

The equalities (31) and (32) do not contain terms with the matrix $\beta_{c}$ .

From Eqs.(5), (21) - (24), (27), (28), we define the operators $\left({H_{CFW}^{\left(+\right)}\left(t\right)}\right)_{odd}$ and $\left({H_{CFW}^{\left(-\right)}\left(t\right)}\right)_{odd}$

[TABLE]

Equations (29), (30) with the expressions (33), (34) and with the use of equalities (25), (26) are reduced to the form

[TABLE]

In (35) and (36), ${\rm{\bf H}}=\mbox{rot}{\rm{\bf A}},\,\,\,{\rm{\bf E}}=-\dfrac{\partial{\rm{\bf A}}}{\partial t}-\nabla A^{0}$ are the magnetic and electric fields.

As a result, we have obtained in the CFW representation the equations for an electron moving in a time-dependent electromagnetic field.

[TABLE]

If we denote

[TABLE]

then we obtain the equations in the Feynman-Gell-Mann representation [4]. In (39) and (40), $\varphi_{FG}({\rm{\bf x}},t),\,\,\chi_{FG}({\rm{\bf x}},t)$ are the upper and lower spinors of the bispinor $\psi_{FG}({\rm{\bf x}},t)$

[TABLE]

From here, the direct connection between the Foldy-Wouthuysen and Feynman-Gell-Mann representationsis evident. Previously [1], a similar connection was established for for the motion of an electron in static electromagnetic fields.

On the other hand, the connections (39) and (40) allow us to solve the problem of ”excess” solutions of Feynman-Gell-Mann equations [4].

In Eqs. (37) and (38), we can perform a similarity transformation and obtain equations using spinor wave functions in the Foldy-Wouthuysen representation.

[TABLE]

For stationary states, the similarity transformation preserves the energy spectrum of a transformed equation. Therefore, the energy spectra of Eqs. (43) and (37) are identical. In this case, the spectra of Eqs. (45) and (38) are also identical. Taking this into account, we can analyze simpler Eqs. (37), (38) with the wave functions in the Feynman-Gell-Mann representation.

5 Conclusions

Using Dirac matrices in the chiral representation, we have for the first time obtained the equations of motion for electrons in the Foldy-Wouthuysen representation in the presence of time-dependent electromagnetic fields. The obtained equations can be used in non-perturbative quantum electrodynamics with strong electromagnetic fields.

In this work, the connection between the Foldy-Wouthuysen and Feynman-Gell-Mann representations has been established in the general case of electromagnetic fields. Previously, such a connection was established in the case of static electromagnetic fields [1].

In the general case of dynamic electromagnetic fields, the wave function in the Foldy-Wouthuysen representation is determined by formulas (19) and (20)

[TABLE]

In the case of static electromagnetic fields,

[TABLE]

The set of basic eigenfunctions $\left({{\begin{array}[]{*{20}c}{\varphi_{c}^{st}({\rm{\bf x}})e^{-i\varepsilon t}\hfill}\\ \,\,\,\,\,\,\,\,\,\,\,0\hfill\\ \end{array}}}\right),\,\,\left({{\begin{array}[]{*{20}c}\,\,\,\,\,\,\,\,\,\,\,0\hfill\\ {\chi_{c}^{st}({\rm{\bf x}})e^{-i\varepsilon t}\hfill}\\ \end{array}}}\right)$ and eigenvalues $\varepsilon_{k}$ includes all the solutions (48) and (49).

The obtained solutions (46) and (47) show how to decompose time-dependent wave functions into basis functions defined by the expressions (48) and (49). The solutions (46), (47) are unambiguously related to the solutions of Feynman-Gell-Mannequations.

6 Acknowledgments

This study was conducted within the framework of the scientific program of the National Center for Physics and Mathematics, section N5 ”Particle physics and cosmology. Stage 2023-2025”.

The author thank A.L.Novoselova for the essential technical assistance in the preparation of the paper.

Bibliography4

The reference list from the paper itself. Each links out to its DOI / PubMed record.

1[1] V. P. Neznamov. Int. J. Mod. Phys. A 40 , 2550049 (2025).
2[2] L. L. Foldy and S. A. Wouthuysen, Phys. Rev. 78 , 29 (1950).
3[3] V. P. Neznamov and A. J. Silenko, J. Math. Phys. 50 , 122302 (2009); arxiv: 0906.2069 (math-ph).
4[4] R. P. Feynman and M. Gell-Mann, Phys. Rev. 109 , 193 (1958).